Methods and apparatus for measuring refractive index and optical absorption differences

ABSTRACT

Methods and instruments are provided for measuring differences in fractional reflectivity changes between transverse electric (TE or s-polarized) and transverse magnetic (TM or p-polarized components of an obliquely incident light with high sensitivity and low noise. Also provided are high sensitivity, low noise methods and instruments for measuring differences in fractional reflectivity changes between R-polarized (right-circularly polarized) and L-polarized (left-circularly polarized) components of a near-normal incident light. The methods take advantage of a nulling step to minimize harmonics of the optical signal derived from a first sample. Determination of odd and even harmonics of the optical signal derived from a second sample allows determination of refractive index and optical absorption coefficient differences between two samples to be determined with high sensitivity and low noise.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.10/272,357, filed Oct. 15, 2002, which claims the benefit of U.S.Provisional Patent Application Ser. No. 60/329,665, filed Oct. 15, 2001,the entire disclosures of which are hereby incorporated by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

Not applicable.

REFERENCE TO A MICROFICHE APPENDIX

Not applicable

BACKGROUND OF THE INVENTION

The invention relates to the field of optics and more particularly tooptical ellipsometry-based methods and apparatus for measuringrefractive index and optical absorption differences.

Optical ellipsometry-based methods for measuring refractive index andoptical absorption of materials are known in the art and have been usedto study a wide-variety of different materials. Prior art ellipsometrymethods, however, tend to lack the signal-to-noise ratios necessary todetect changes in optical response from a few percent of one monoatomicor monomolecular layer, thus in general restricting their use to systemsgiving rise to changes greater or much greater than a few percent of onemonoatomic or monomolecular layer. In cases of special forms ofellipsometry such as oblique-incidence optical reflectance differenceand surface photo-absorption, prior art ellipsometry techniques teachthe measurement of only one or one combination of the refractive indexand the optical absorption coefficient rather than two, and so fail torealize the benefits of simultaneous determinations of refractive indexand optical absorption coefficient differences. Prior art ellipsometrytechniques also fail to provide a solution to the problem of determininga refractive index difference or an optical absorption coefficientdifference between two samples under circumstances in which thesignal-to-noise ratios of the first or the second order harmonic isinadequate because of environmental or instrument noise in these regionsof the spectrum.

The present invention addresses these and other deficiencies of theprior art by providing methods and apparatus for determining refractiveindex and optical absorption coefficient differences with significantlyimproved signal-to-noise properties, for simultaneous monitoring ofrefractive index and optical absorption coefficient differences, and forobtaining higher order harmonic measurements.

BRIEF SUMMARY OF THE INVENTION

In one aspect, the invention provides for a method of determining arefractive index difference between two samples, the method comprisingilluminating a first sample at an oblique incidence angle with apolarized incident light beam, said first sample reflecting at leastsome of said incident light beam to form a reflected light beam;

-   -   modulating at a modulation frequency the polarization of at        least one of said incident light beam and said reflected light        beam;    -   nulling said reflected light beam by interacting said incident        light beam or said reflected light beam with a phase shifter and        adjusting a phase difference between an s-polarized component        and a p-polarized component of said reflected light beam,        wherein said adjustment minimizes an odd modulation frequency        harmonic of said reflected light beam intensity;    -   without further adjustment of said phase difference,        illuminating a second sample at said oblique incidence angle        with said incident light beam, said second sample reflecting at        least some of said incident light beam to form a second        reflected light beam; and    -   determining a magnitude of an odd modulation frequency harmonic        of said second reflected light beam intensity, wherein said        magnitude corresponds to a refractive index difference between        said first and said second samples.

In another aspect, the invention provides for a method of determining arefractive index difference between two samples, the method comprisingilluminating a first sample at a near-normal incidence angle with anincident light beam, said first sample reflecting at least some of saidincident light beam to form a reflected light beam;

-   -   modulating at a modulation frequency the polarization of at        least one of said incident light beam and said reflected light        beam;    -   nulling said reflected light beam by interacting said incident        light beam or said reflected light beam with a phase shifter for        circularly-polarized light and adjusting a phase difference        between a right-circularly polarized component and a        left-circularly polarized component of said reflected light        beam, wherein said adjustment minimizes an odd modulation        frequency harmonic of said reflected light beam intensity;    -   without further adjustment of said phase difference,        illuminating a second sample at said near-normal incidence angle        with said incident light beam, said second sample reflecting at        least some of said incident light beam to form a second        reflected light beam; and    -   determining a magnitude of an odd modulation frequency harmonic        of said second reflected light beam intensity, wherein said        magnitude corresponds to a refractive index difference between        said first and said second samples.

In yet another aspect, the invention provides for a method ofdetermining an optical absorption coefficient difference between twosamples, the method comprising illuminating a first sample at an obliqueincidence angle with a polarized incident light beam, said first samplereflecting at least some of said incident light beam to form a reflectedlight beam;

-   -   modulating at a modulation frequency the polarization of at        least one of said incident light beam and said reflected light        beam;    -   nulling said reflected light beam by interacting said incident        light beam or said reflected light beam with an analyzer and        adjusting the analyzer transmission axis, θ_(PL), wherein said        adjustment minimizes an even modulation frequency harmonic of        said reflected light beam intensity;    -   without further adjustment of θ_(PL), illuminating a second        sample at said oblique incidence angle with said incident light        beam, said second sample reflecting at least some of said        incident light beam to form a second reflected light beam; and    -   determining a magnitude of an even modulation frequency harmonic        of said second reflected light beam intensity, wherein said        magnitude corresponds to an optical absorption coefficient        difference between said first and said second samples.

In still another aspect, the invention provides for a method ofdetermining an optical absorption coefficient difference between twosamples, the method comprising illuminating a first sample at anear-normal incidence angle with an incident light beam having aright-circularly polarized component and a left-circularly polarizedcomponent, said first sample reflecting at least some of said incidentlight beam to form a reflected light beam;

-   -   modulating at a modulation frequency the polarization of at        least one of said incident light beam and said reflected light        beam;    -   nulling said reflected light beam by interacting said incident        light beam or said reflected light beam with an analyzer and        adjusting the analyzer transmission axis, wherein said        adjustment minimizes an even modulation frequency harmonic of        said reflected light beam intensity;    -   without further adjustment of the analyzer, illuminating a        second sample at said near-normal incidence angle with said        incident light beam, said second sample reflecting at least some        of said incident light beam to form a second reflected light        beam; and    -   determining a magnitude of an even modulation frequency harmonic        of said second reflected light beam intensity, wherein said        magnitude corresponds to an optical absorption coefficient        difference between said first and said second samples.

In one aspect of the invention, the first sample corresponds to a knowncontrol sample, and the second sample corresponds to an unknown sample.In another aspect, the known control sample is a blank sample. In yetanother aspect, the first and second samples are located in discretespatial regions, while in another aspect, the first and second samplesare located in the same spatial region and the measurements of saidfirst and said second samples are taken at different time points.

In another aspect, the invention provides for devices for determining arefractive index difference, an optical absorption coefficientdifference, or both refractive index difference and optical absorptioncoefficient difference between two samples according to one or more ofthe above methods, said device including components adapted to carryingout one or more of the above method steps.

These and other advantages of the present invention will now bedescribed with reference to the drawings and written description whichare intended to exemplify but not limit the invention. Variations anddepartures from the exemplified embodiments will be obvious to personsof skill in the art and are intended to be within the spirit of theinvention, the scope of which is to be limited only by the claims. Allreferences to patents, publications and other materials are herebyincorporated by reference in their entirety for all purposes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 diagrams an example of an instrument for carrying out the methodsof the invention.

FIG. 2 illustrates an image scan using the first harmonics over an areaof a hybridized gene chip.

FIGS. 3(a)-3(d) illustrate scans over a region of a microarraycontaining 5×5 (=25) printed features of 60-mer DNA oligonucleotideshybridized with Cy5-labeled cDNA. FIG. 3(a) shows Im{Δ_(p)−Δ_(s)}. FIG.3(b) shows Re{Δ_(p)−Δ_(s)} for same region as shown in FIG. 3(a). FIG.3(c) shows Im{Δ_(p)−Δ_(s)} for another region of same microarray asshown in FIGS. 3(a) and 3(b), but where none of the Cy5-labeled cDNAhybridized. FIG. 3(d) shows Re{Δ_(p)−Δ_(s)} for same region as shown inFIG. 3(c).

FIG. 4 illustrates Im{Δ_(p)−Δ_(s)} during continuous growth of xenonmonolayers on Nb(110) at different temperatures.

FIG. 5. illustrates Re{Δ_(p)−Δ_(s)} during continuous growth of xenonmonolayers on Nb(110) at 40 K.

FIGS. 6(a)-6(b) respectively illustrate Re{Δ_(p)−Δ_(s)} andIm{Δ_(p)−Δ_(s)} during electrodeposition of Pb monolayers on Cu(100) asa function of applied potential and time.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Definitions

All terms, unless specifically defined below, are intended to have theirordinary meanings as understood by those of skill in the art. Masses andvolumes are intended to encompass variations in the stated quantitiescompatible with the practice of the invention. Such variations arecontemplated to be within, e.g. about ±10-20 percent of the statedquantities. In case of conflict between the specific definitionscontained in this section and the ordinary meanings as understood bythose of skill in the art, the definitions supplied below are tocontrol.

“Interacting” a light beam with a component encompasses all manner ofreflection, transmission, scattering, and combinations thereof. Theparticular modality of interaction is determined by the choice of thecomponent and the desired effect on the properties of the light beambrought about by the interaction.

“Adjusting” an analyzer or a polarizer refers to changing theorientation of the analyzer or the polarizer.

A “biological molecule” encompasses any molecule synthesized or foundwithin a biological cell including by way of example but not limitation,a protein, a nucleic acid, a carbohydrate, and a lipid.

An “incident light beam” refers to a light beam that has not reflectedoff a sample.

A “reflected light beam” refers to a light beam that has reflected off asample.

The optical ellipsometry techniques of the present invention provideways to measure differences in the refractive index and opticalabsorption coefficients of two samples. An advantage of the presentinvention over the prior art is that the methods of the inventioninclude one or more nulling steps that render the methods sufficientlysensitive to measure differences between samples corresponding to theaddition, removal, replacement or alteration of as small as a fewpercent or less of the atoms or molecules in a monoatomic ormonomolecular layer of a sample or equivalent changes. The methods ofthe invention also allow for the detection of quite subtle structural,electronic, magnetic, or other alterations that affect a substantialportion of the atoms or molecules present in a monoatomic ormonomolecular layer of a sample.

Thus, the methods are sufficiently sensitive to detect, e.g., molecularbinding events such as nucleic acid hybridizations, protein-proteininteractions, protein-nucleic acid interactions, conformation changes inmacromolecular structures, and the addition, deletion, replacement oralteration of a few percent of a monolayer mass to the surface of asample. The methods thus find broad applicability to monitoringbiochemical and other binding events and in some instances cansupplement or replace monitoring techniques that use specialized labelssuch as fluorophores to follow these events. While these advantages ofthe invention have been described with reference to alterationsoccurring within monoatomic or monomolecular layers of a sample, themethods of the invention are equally applicable to bulk samples.Similarly, while the advantages of the present invention have beendescribed with reference to biological macromolecules such as nucleicacids, proteins, etc., the methods are applicable to the study ofnon-biological materials such as organic and inorganic crystal growth,corrosion, interfacial chemical reactions, electrochemical deposition,etc.

The samples may comprise any material whatsoever provided the samplematerials interact with light to produce a change in the fractionalreflectivity between polarized components of light interacting with thesamples. Thus the samples may comprise non-chiral and non-magneticmaterials or materials whose chiral and magneto-optic effects can beneglected for the application of interest. For these types of samples(Class I samples), the method preferably is practiced usingoblique-incidence geometry. In other embodiments, the samples maycomprise chiral or magnetic materials whose chiral or magneto-opticproperties are of interest. For these types of samples (Class IIsamples), the method preferably is practiced using normal or near-normalincidence geometry. In a preferred embodiment, the two samples comprisetwo different spatial regions such as two different regions of an arrayof features, such as, e.g., an array of nucleic acids or an array ofproteins, or any other type of array, although the methods of theinvention are not limited to arrays. We refer to these as samples thatare displaced in space. A first sample may comprise a blank or controlregion lacking a material of interest, while a second sample maycomprise a region comprising a material of interest that an investigatorwishes to detect or characterize. Of course, the method is equallyapplicable to two different regions both having materials of interest.

In another preferred embodiment, the two samples comprise the samespatial region that differs in composition as a function of, e.g., time.Thus measurements may be made from the same region at two differentpoints in time, to compare the optical properties of the two samplesaccording to the methods of the invention. We refer to these as samplesthat are displaced in time. In this case, the methods of the inventionare used to determine whether there has been a change in the opticalproperties of that region over time. Such changes can be indicative of,e.g., an alteration of protein conformation, a molecular binding event,such as a nucleic acid hybridization event or a protein-protein orprotein-nucleic acid binding event, or a molecular dissociation eventindicating the disruption of a molecular complex. Thus, the methods ofthe invention are useful for following biochemical reactions, includingreactions carried out using small quantities of materials such as thequantities bound to microarrays.

Using a gene chip as an example of Class I samples, single-strandedsegments of “reference” nucleic acids such as deoxyribonucleic acids(DNAs) or ribonucleic acids (RNAs) can be deposited or fabricated on asubstrate in the form of a two-dimensional array of “features” orpatches. Subsequently, these “reference” nucleic acids are allowed tohybridize with single-stranded segments of a sample nucleic acid such asa sample DNA or RNA. By labeling the sample DNA with one or morefluorescent dyes or fluorophores, the binding affinity or the degree ofhybridization can be read out in the form of a fluorescence yield usinga confocal fluorescence microscope.

Although fluorescence labeling is widely used, the process is not alwaysefficient or specific or non-intrusive enough to ensure adequatesensitivity, contrast, and fidelity. For example, photo-bleaching by theilluminating light or ambient light can render fluorescence-labeledsamples and/or hybridized chips difficult to handle. Photobleaching alsocan make it impractical to reexamine features on a gene chip.Furthermore, variations may occur during deposition or in situfabrication of the “reference” nucleic acid and in the subsequenthybridization process. As a result, a straightforward interpretation ofthe fluorescence yield as an indicator of the binding affinity of thesample nucleic acid or extent of hybridization may not always bepossible. These difficulties may be overcome using the methods of theinvention insofar as the methods may be practiced using unlabeledmaterials or using stable, non-fluorescent labels.

We refer to the methods of the invention used in conjunction with ClassI samples as an oblique-incidence optical reflectivity differencetechnique (OI-RD), that (1) complements and in some cases replacesscanning fluorescence microscopes; (2) has multiple contrast modes thatcan be used to verify the extent of molecular binding or dissociationevents such as, e.g., nucleic acid hybridization, protein/nucleic acidinteractions, or protein-protein interactions; and (3) has thesensitivity to detect changes in a sample as small as a few percent ofone monoatomic or monomolecular layer of a sample.

In one embodiment, the methods of the invention measure the differencein fractional reflectivity change between the transverse electric (“TE”or “s-polarized”) and transverse magnetic (“TM” or “p-polarized”)components of an obliquely incident light. By obliquely incident, wemean incident light that is not normal nor near-normal to the plane of asurface comprising a sample. The essence of OI-RD is as follows.

Let r_(p0)=|r_(p0)| exp(iΦ_(p0)) and r_(s0)=|r_(s0)| exp(iΦ_(s0)) be thereflection coefficients for p- and s-polarized light from a substratesurface having a first sample. In one preferred embodiment this firstsample is a control sample that is known to lack a material of interest,e.g., the sample consists of the substrate only and not the material ofinterest. Let r_(p)=|r_(p)|exp(iΦ_(p)) and r_(s)=|r_(s)| exp(Φ_(s)) bethe reflection coefficients for p- and s-polarized light from asubstrate surface having a second sample. In this preferred embodiment,the second sample comprises the substrate and a material of interest.

We measure the fractional reflectivity change Δr/r₀=(r−r₀)/r₀ in the p-and s-polarized components of a light impinging on the sample at anoblique angle of incidence. We define S_(N) as the magnitude of thes-polarized component and P_(N) as the magnitude of the p-polarizedcomponent. These magnitudes will be used below in describing conditionsthat must be met to practice this and other embodiments of theinvention. In a preferred embodiment, oblique-incidence confocal imagingoptics such as are shown in FIG. 1 are used, although any system havingoptical components capable of providing adequate spatial resolution andbrightness may be used in accordance with the methods of the invention.In this preferred embodiment, a collimated monochromatic ormonochromatized light source is used, although it is not crucial thatmonochromatic or monochromatized light be used. What is required isthat, as explained below, the phase lag, φ, can be adjusted so that anodd harmonics of the reflected light is minimized when the first sampleis being illuminated. Thus, if a broad band phase-shifter such as aFresnel rhomb achromatic retarder (Thermo Electron Corporation, Waltham,Mass.) with variable incidence angle is used, the requirement formonochromaticity may be relaxed completely.

In a preferred embodiment, the incident light beam with intensityI_(inc) is initially p-polarized or s-polarized. The general conditionthat must be met, however is that the incident light beam has an initiallinear polarization with S₁≠P₁. The polarization of the beam is alteredat a frequency Ω with a polarization modulator such as a photoelasticmodulator, (e.g., model PEM-90, Hinds Instruments, Inc. Hillsboro,Oreg.), or an electro-optic phase modulator, (e.g., models 4001, 4002,4003, 4004, 4421, 423, 4431, or 4433 available from New Focus, Inc. SanJose, Calif.), or any other device that will alter the polarization fromthe initial linear polarization to a combination of the initial linearpolarization and a polarization that includes an orthogonal componentmissing from the initial linear polarization.

For example, if the light beam is initially p-polarized, then thepolarization modulator is used to add an s-polarization component to thebeam, and to switch at frequency Ω between the initial p-polarized lightand light having an s-polarization component. Similarly, if the lightbeam is initially s-polarized, then the polarization modulator is usedto add a p-polarized component to the beam, and to switch at frequency Ωbetween the initial s-polarized light and the light having a p-polarizedcomponent. Let S₂ and P₂ be the magnitudes of the s-polarization andp-polarization components of the polarization-modulated beam. Thegeneral condition that must be met for the polarization-modulated beamis that $\frac{S_{1}}{P_{1}} \neq {\frac{S_{2}}{P_{2}}.}$

The relative amounts of s-polarization and p-polarization componentspresent in the initial linear polarization and in the modulatedpolarization may vary depending on the application for which the methodis used. What is required is that the S₁≠P₁ and$\frac{S_{1}}{P_{1}} \neq \frac{S_{2}}{P_{2}}$conditions set forth above are chosen to yield adequate signal-to-noiseratios in the even and odd harmonics of the reflected light, asdescribed below. For example, an initially p-polarized beam (S₁=0, P₁≠0)may be switched by a modulator completely to an s-polarized beam (S₂≠0,P₂=0) or be switched to a nearly p-polarized beam with a smalls-polarization component (P₁≈P₂ but P₂>>S₂). This will, of course, varydepending on, e.g., the instrumentation used for detection of thereflected light, and the nature of the samples under investigation, andcan be readily determined by an ordinarily skilled person.

Polarization modulation may be done in a continuous or a discretemanner. By continuous modulation, we mean the change in polarization iscontinuous in time, such as that produced by a photo-elastic modulator,an electro-optic modulator, or a wave plate on a continuously rotatingwheel. Discrete modulation refers to use of a discrete set of mixedpolarizations chosen to obtain the even and odd “harmonics”algebraically, as described in greater detail below.

The polarization-modulated beam is then passed through a phase shiftingdevice such as a voltage-controlled Pockel cell (Cleveland Crystals,Highland Heights, Ohio), or a mechanically controlled Soleil-Babinetcompensator (Coherent Inc., Auburn, Calif.; Cleveland Crystals Inc.,Highland Heights, Ohio), or a Berek compensator (New Focus, San Jose,Calif.), or a double-Fresnel rhomb achromatic retarder (Thermo Oriel,Stratford, Conn.; Halbo Optics, Chelmsford, UK) so that a variable phaselag, φ, can be added to the s-polarized component, relative to thep-component. The beam is subsequently focused on the first samplesurface at an oblique incident angle θ_(inc). The reflected beam isrecollimated and sent through an analyzing polarizer (also referred toas a polarization analyzer or analyzer) such as, e.g., a Glan laserdouble escape window polarizer, model CPAD (CVI Laser, Albuquerque, N.Mex.) or Glan-Thompson polarizer model 5524 (New Focus, San Jose,Calif.), the analyzer having its transmission axis set at angle θ_(PL)from the s-polarization axis.

The resulting intensity of the reflected, analyzed beam, I_(R)(t) isdetected using, e.g., a photomultiplier, or a diode based photodetectorsuch as a biased silicon photodiode, or any other device capable oftransducing light intensity into a variable electrical signal. I_(R)(t)has a time invariant component I_(R)(0) that is the mean intensity ofthe reflected light, as well as terms that vary with time at variousharmonics of the modulation frequency Ω.

The odd and even harmonics I_(R)(NΩ)cos[(NΩ)t] for N=odd and N=even,respectively are detected by analyzing the signal output from the lighttransducer device using a lock-in amplifier such as a NationalInstrument Dynamic Signal Acquisition Module (model NI-4472 for PCI orPXI) controlled with for example National Instruments LabVIEW Softwareand the Lock-In Amplifier Start-up Kit available from NationalInstruments (Austin, Tex.), or models SR810, or SR830, or SR844, orSR850, or SR530, or SR510 available from Stanford Research Systems(Sunnyvale, Calif.), or models 7280, or 7265, or 7225, or 7225BFP, or7220BFP, or 5105, or 5106, or 51 1A, or 5209, or 5210 available fromPerkin Elmer Instruments (Oak Ridge, Tenn.), or a spectrum analyzer suchas an ESA-L series portable spectrum analyzer, or an ESA-E seriesportable spectrum analyzer, or an 8560E series portable spectrumanalyzer, or an 8590 series portable spectrum analyzer or a PSA serieshigh-performance spectrum analyzer, or an MMS modular measurementsystem, or swept-tuned spectrum analyzer available from Agilent (PaloAlto, Calif.), or any other device or method capable of detecting withsufficient signal-to-noise ratios spectral components of the varyingsignal output from the light transducer device. For example, theharmonics can be obtained by digitizing the signal, such as, e.g., aphotodiode or photomultiplier current, loading the digitized signal intoa computer, Fourier transforming the digitized signal using, e.g., anFFT algorithm, to obtain the power spectrum of the signal, anddetermining the power of a frequency component corresponding to an oddor even harmonic of the modulation frequency, Ω.

One of the key aspects of the invention is that when the incident beamilluminates the first sample, θ_(PL) is adjusted such that the intensityof an even harmonic of the reflected analyzed beam, I_(R)(NΩ), isminimized (N=even), and the phase lag φ is adjusted such that theintensity of an odd harmonic of the reflected analyzed beam, I_(R)(N′Ω), is minimized (N′=odd). We refer to these minimization steps as“nulling” steps. In a preferred embodiment, N=even=2, and θ_(PL) isadjusted such that I_(R)(2Ω)=0, and N′=odd=1 and φ is adjusted such thatI_(R) (Ω)=0. In this preferred embodiment, for substrates that aretransparent or minimally absorptive, when the incident beam illuminatesa second sample, the subsequent changes in I_(R)(Ω) and I_(R)(2Ω) ariserespectively from differences in refractive index and optical absorptioncoefficient between the first and second samples.

Thus, if the first sample consists of a transparent substrate only andnot a material of interest, such as a glass surface, and the secondsample comprises a material of interest, e.g., a region of nucleic acidon the glass substrate, then the changes in I_(R)(NΩ) for N=odd andN=even between the two samples come from the change in refractive indexand optical absorption coefficient from those of the substrate to thoseof the nucleic acid, or whatever substance is bound or adsorbed to thesubstrate. The refractive index and optical absorption coefficient ofthe substance are derived from Δ_(p)−Δ_(s) whereΔ_(p)=(r_(p)−r_(p0))/r_(p0) and Δ_(s)=(r_(s)−r_(s0))/r_(s0). The realand imaginary parts of Δ_(p)−Δ_(s) are determined from the followingmeasurement,I _(R)(NΩ)≅c ₁(N)I _(inc) |r _(p0) sin θ _(PL)|² Im{Δ _(p)−Δ_(s)} forN=odd   (1)I _(R)(NΩ)≅c ₂(N)I _(inc) |r _(p0) sin θ_(PL)|² Re{Δ _(p)−Δ_(s)} forN=even   (2)c₁(N) and c₂(N) are two unit-less numerical constants that depend on Nand how the initial polarization is modulated. It can be shown that on atransparent substrate such as a glass slide, the odd harmonics, I_(R)(NΩ) for N=odd, or Im{Δ_(p)−Δ_(s)} is a measure of the variation ofrefractive index, while the even harmonics, I_(R) (NΩ) for N=even, orRe{Δ_(p)−Δ_(s)} is a measure of the variation of the optical absorptioncoefficient.

The real and imaginary parts of Δ_(p)−Δ_(s) characterize the material ofinterest relative to the substrate which is used to null the harmonics.For this reason, it may be advantageous to eliminate the systemdependent factors c₁(N)I_(inc)|r_(p0) sin θ_(PL)|² andc₂(N)I_(inc)|r_(p0) sin θ_(PL)|² in front of (Δ_(p)−Δ_(s)). To obtainthe factor in front of lm{Δ_(p)−Δ_(s)}, the beam is focused on the firstsample and the phase lag φ is adjusted so that the detected beamintensity at the odd harmonic is maximized, and the maximum strength ofthe detected signal so obtained is equal to c₁(N)I_(inc)|r_(p0) sinθ_(PL)|². To obtain the factor in front of Re{Δ_(p)−Δ_(s)}, the evenharmonic is nulled in the region of the first sample, and a secondanalyzer having its transmission axis oriented along either thep-polarization or s-polarization axis is inserted in the light pathimmediately before or immediately after the sample. For the embodimentillustrated in FIG. 1, this corresponds to inserting the second analyzerinto the light path in a region between either of the two lenses 106 and107 and the sample. The detected beam intensity at the even harmonic isequal to one half of c₂(N)I_(inc)|r_(p0) sin θ_(PL)|², multiplied by thetransmittance of the second polarizer for the passing polarizationcomponent. There are other methods to obtain these two factors, as willbe recognized by those of ordinary skill in the art.

Note that if the material of interest is on a substrate that is nottransparent, i.e., a substrate that absorbs light, the refractive indexand the optical absorption coefficients of the material of interest arenot separately determined from the odd and even harmonics of thedetected light intensity as described above. Rather, algebraic methodsmust be used to determine these two parameters using both odd and evenharmonics information, as set forth in the attached appendix.

In carrying out the methods of the invention, one of ordinary skill willreadily appreciate how to select a harmonic that provides adequatesignal-to-noise properties. While in principle, all odd harmonicsprovide equivalent information about variation in refractive index, andall even harmonics provide equivalent information about variation inoptical absorption coefficient, it can fortuitously arise that e.g., thefirst harmonic is in a noisy region of the spectrum because of, e.g.,instrument or environment noise, and so the third or fifth orhigher-order odd harmonic should be obtained for purposes of evaluatingrefractive index variations.

The relative signal strengths of various harmonics can be adjusted, ifnecessary, by, e.g., varying the depth of polarization modulation. Inmost cases, the values of the first through seventh order odd harmonicscan be made to have values within a factor of two of each other, andsimilarly, the values of the second through eighth order even harmonicsalso can be made to have values within a factor of two of each other.This offers sufficient flexibility for most practical applications ofthe methods of the invention.

As one of skill in the art readily will appreciate, subsequent sampleregions can be interrogated using the above-described techniques, andthe odd and even harmonics will show the differences in the refractiveindex and optical absorption coefficient between the first andsubsequently interrogated sample regions.

If a sample is labeled with a dye and monochromatic light having awavelength within the dye's absorption band is used in the method, thena scan of the sample to determine an even harmonic as a function oflocation will provide an indication of the relative amounts of dye inthe interrogated locations. For purposes of illustration, consider anucleic acid array hybridized with a fluorescent-labeled nucleic acid.According to the methods of the invention, a scan of the array surfaceto obtain an even harmonic using a wavelength corresponding to theabsorption band for the fluorescent dye will provide information roughlyequivalent to a fluorescence yield scan of the array obtained using afluorescence microscope. That is, the even harmonic will provideinformation that corresponds to the location and amount of labelednucleic acid present on the array. Compared to fluorescence microscopy,though, the methods of the invention have the advantage of not relyingon fluorescence yield of a fluorophore which can vary as a result ofquenching or photobleaching phenomena, but instead on the absorptionproperties of the dye, which tend to be more stable than the emissionproperties.

For purposes of illustration, consider the same type of array hybridizedwith an unlabeled nucleic acid sample and illuminated with light at awavelength other than an absorption band wavelength for the nucleicacid. A scan of the array surface according to the methods of theinvention to obtain an even harmonic will not detect features on thearray surface. However, a scan of the array surface to obtain an oddharmonic will show variations in refractive index along the surface ofthe array. This variation corresponds to the surface density of nucleicacid along the array surface, and so provides a convenient way tomeasure, e.g., hybridization of even an unlabeled nucleic acid to thearray. See Appendix Special Case 4. Similarly, an odd harmonic scan canbe used to examine the array prior to hybridization to ensure thequality of the array. Because the method is non-destructive, an arraycan be repeatedly scanned using the methods of the invention. This is incontrast to fluorescence-based methods in which repeated scans can leadto signal loss due to photo-bleaching.

The embodiment illustrated in FIG. 1 makes use of spherical lenses (106,108) for the focusing optics so that a small spatial region i.e., aspot, is illuminated and the reflected light from the illuminated regionis analyzed to obtain one or more of the harmonics. In someapplications, such as those involving the use of two-dimensional arrays,it is advantageous to analyze a line of features on the array at once,rather than sequentially analyzing each feature within the line. Theadvantage in time saving can be as much as N-fold, where N is the numberof spots within a line. For high-density oligonucleotide arrays orchips, the amount time required to analyze the features on the chip canbe reduced by as much as 1000-fold using line analysis in lieu of spotanalysis. Line analysis can be carried out using cylindrical lenses asthe focusing optics (106) on the illumination side. On the reflectionside, spherical optics (108) may be used to image the reflected beamonto a linear detector array (110) such as a linear diode array insteadof a single detector.

It also is possible to use a two-dimensional (2D) detector array such asa charge-coupled device (CCD) detector or a 2D image intensifier toanalyze a two-dimensional array of features simultaneously. A twodimensional area is illuminated and the light reflected from theilluminated area is focused on the two-dimensional detector to form animage of the illuminated area. Spectral analysis of the signals arisingfrom the discrete regions of the two dimensional detector is carried outaccording to the methods of the invention to obtain the desiredharmonics.

In addition to the methods described above, other techniques can be usedby one of ordinary skill to practice the invention. For example, thepolarization modulation has been described above by reference toelectro-optic and photo-elastic modulation. In addition, one of ordinaryskill will appreciate that other methods can be used for polarizationmodulation such as, e.g., mechanical rotation of a wave plate having anyarbitrary phase retardation between two orthogonal polarizations, or adouble-Fresnel rhomb mounted on a spinning wheel or rotation stage.Polarization modulation also can be carried out by reflecting the beamoff one or more sets of surfaces or using a combination of transmissionand reflection optics. An example of using a reflection geometry toobtain the polarization modulation is use of a double-Fresnel rhomb orone or a pair of reflecting surfaces mounted on a spinning wheel.

Another embodiment of the method also can be practiced using the samesetup as described above with the variation that the phase-shifter isplaced after the sample, and before the analyzer. The requirements thatS₁≠P₁ and $\frac{S_{1}}{P_{1}} \neq \frac{S_{2}}{P_{2}}$pertain to this embodiment as well.

In lieu of polarization modulation of the incident beam, thepolarization modulation can be introduced after the beam has beenreflected off a sample surface. In this embodiment, the arrangement ofthe optics is reversed with respect to what is shown in the embodimentillustrated in FIG. 1. Thus, a light beam is passed through an analyzerhaving its transmission axis oriented at θ_(PL) with respect to thes-polarization axis; the resultant beam is used to illuminate the samplesurface at an oblique incidence angle θ_(inc), and forms a reflectedbeam; the reflected beam then is passed through a phase-shifter such as,e.g., a Pockel cell to introduce a phase lag between the s- andp-polarized components of the reflected beam; the reflected beam towhich the phase lag has been added is then polarization modulated bypassing the phase-lagged, reflected beam through a device such as anelectro-optic or photo-elastic modulator, or a rotating wave plate, or adouble Fresnel rhomb mounted on a spinning wheel or rotation stage asdescribed above so that the polarization of the beam is periodically ordiscretely modulated.

The resultant beam then is passed through a polarizer that passes anunequal combination of p-polarization and s-polarization components oris passed through or reflected off any device so that either of the s-or p-polarized components or an unequal combination of the s- andp-polarized components of the beam is detected by a detector. In thisway, the odd harmonic after nulling by adjustment of the phase lag willmeasure Im{Δ_(p)−Δ_(s)}, and the even harmonic after nulling byadjustment of the analyzer's transmission axis, namely, θ_(PL), willmeasure Re{Δ_(p)−Δ_(s)} between two samples displaced in space or time.

The conditions for embodiments in which the polarization modulation isintroduced after the beam has interacted with the sample are as follows.The initial beam need not but may be linearly polarized. An arbitraryinitial polarization is sufficient. It also is required that asufficient amount of light pass through the analyzer to provide adequatesignal-to-noise ratios for practice of the invention. As discussedabove, the amount of light required to provide adequate signal-to-noiseratios will vary depending upon the details of the sample and systemcomponents used, and can readily be determined by one of ordinary skillin the art. Once the beam passes through the phase-shifter, it emergeswith s-polarization and p-polarization component magnitudes S₁ and P₁.After passing through the polarization modulator, the beam emerges withs-polarization and p-polarization component magnitudes S₂ and P₂. Thenecessary condition is that$\frac{S_{1}}{P_{1}} \neq {\frac{S_{2}}{P_{2}}.}$It also is a necessary condition that the polarizer's transmission axisis not oriented at ±45° with respect to the s-polarization axis.

In another variation of methods in which polarization modulation isintroduced after the beam has been reflected off a sample surface, thephase-shifter is placed between the analyzer and the sample, and thes-polarization and p-polarization component magnitudes S₁ and P₁ arecharacteristic of the beam after it has interacted with the sample.

Two additional embodiments employing an initially elliptically polarizedbeam also are within the scope of the present invention. In the first“elliptically polarized” embodiment, the setup comprises, in order, alight beam with an initial elliptical polarization (preferably thesource of this beam is a laser), a device that variably changes therelative magnitude of the s-polarization component to the p-polarizationcomponent (serving as the analyzer) (e.g., a wave plate, or a doubleFresnel rhomb, or a mirror, or a set of mirrors, or a transparentmaterial, or a set of transparent materials), sample, phase-shifterthrough which the beam emerges with s-polarization and p-polarizationcomponent magnitudes S₁ and P₁, polarization modulator through which thebeam emerges with s-polarization and p-polarization component magnitudesS₂ and P₂ with the necessary condition that${\frac{S_{1}}{P_{1}} \neq \frac{S_{2}}{P_{2}}},$polarizer with its transmission axis not oriented at ±45° with respectto the s-polarization axis, and detector. The second “ellipticallypolarized” embodiment differs from the first “elliptically polarized”embodiment in that the positions of the sample and phase-shifter arereversed and that the s-polarization and p-polarization componentmagnitudes S₁ and P₁ are characteristic of the beam after it hasinteracted with the sample.

To summarize, the methods of the invention for use with Class Imaterials have been described above with respect to six differentoptical setups:

-   -   Setup 1: light source with initial linear polarization,        polarization modulator, phase-shifter, sample, analyzer,        detector    -   Setup 2: light source with initial linear polarization,        polarization modulator, sample, phase-shifter, analyzer,        detector    -   Setup 3: light source with arbitrary initial polarization,        analyzer, sample, phase-shifter, polarization modulator,        polarizer, detector    -   Setup 4: light source with arbitrary initial polarization,        analyzer, phase-shifter, sample, polarization modulator,        polarizer, detector    -   Setup 5: light source with initial elliptical polarization,        analyzer, sample, phase-shifter, polarization modulator,        polarizer, detector    -   Setup 6: light source with initial elliptical polarization,        analyzer, phase-shifter, sample, polarization modulator,        polarizer, detector

Nulling of the first order or odd harmonics has been described above byreference to electro-optic devices such as a Pockel cell or variablewave plates such as a Babinet compensator, a Soleil compensator, or aBerek compensator. However, one of ordinary skill can carry out thefirst order or other odd harmonic nulling step using a device based uponphoto-elastic effect such as a mechanically driven photoelasticmodulator or a tilted uniaxial or bi-axial wave plate such that theoptical path for the two orthogonal polarized components (i.e. thep-polarization and the s-polarization components) can be changed toobtain the desired phase lag.

If the second or other even order harmonic is not required for purposesof the analysis being carried out according to the methods of theinvention (i.e., if optical absorption coefficient variations are not tobe obtained), then the nulling step for the second or other even orderharmonic can be skipped, and instead, the transmission axis of theanalyzer can be set to any angle θ_(PL) other than the two orthogonalangles that allow transmission only of the p-polarized or thes-polarized components. For practicing the embodiments of the inventionthat do not require determination of even order harmonics, the analyzercan be replaced with any optical device (such as a tilted, transparentoptical window in the beam path such that the incidence plane to thewindow bisects the s-polarization and the p-polarization axes) or otheroptical arrangements that will mix the s-polarized and the p-polarizedcomponents before the beam reaches the detector.

Second or other even order harmonics also may be nulled using methodsother than adjustment of the transmission axis of the analyzer asdescribed above. For example, any optical device that can change therelative amplitudes of the s-polarized and the p-polarized components ofa light beam, either in transmission mode, in which the light beampasses through the device, or in reflectance mode, in which the lightbeam is reflected off one or a set of surfaces having differentreflectivities for the s-polarized and the p-polarized components may beused. An example of this is provided in A. Wong and X. D. Zhu, “Anoptical differential reflectance study of adsorption and desorption ofXenon and Deuterium on Ni(111),” Appl. Phys. A 63, 1 (1996), which ishereby incorporated by reference in its entirety for all purposes.

The periodic polarization modulation may also be eliminated altogetherand one can make measurements using a discrete set of mixedpolarizations chosen to obtain the even and odd “harmonics”algebraically. This may be advantageous for imaging one line at a timeor the entire area of the interrogated sample surface with a low dcnoise detector such as a cooled CCD. In this case, for example, one canpass the initial light beam through, say, a quarter-wave plate, or anywave plate other than a half-wave plate. Let φ be the angle between theinitial linear polarization and one of the principal axes of the waveplate. One can measure the light intensity after the analyzer at φ=0,π/4, and 3π/4. Then, for transparent substrates or substrates with lowlight absorption, [I_(R)(φ=7π/4)−I_(R)(φ=π3/4)] is proportional tolm{Δ_(p)−Δ_(s)} after nulling with a phase shifting device such a Pockelcell following the above-described procedure or variations of it; while[2I_(R)(φ=0)−I_(R)(φ=π/4)−I_(R)(φ=3π/4)] is proportional toRe{Δ_(p)−Δ_(s)} after nulling with the analyzing polarizer following theabove-described procedure or variations of it.

A group of setups and constraints analogous to those described above maybe used to practice embodiments of the invention with Class IImaterials, i.e., chiral or magnetic materials whose chiral ormagneto-optic properties are of interest. For these setups, in lieu ofan oblique incidence geometry, near-normal incidence geometry is used. Aperfectly normal incidence geometry is not always practical as theillumination section of the setup will more often than not interferewith the detection section (see FIG. 1). Usually one way to overcome theproblem is to use a near-normal incidence geometry. The criterion isthat the contribution from the non-magnetic and non-chiral part of thematerial response through the now non-vanishing Δ_(p)−Δ_(s) term iseither small compared to the chiral or magneto-optic part of theresponse, or can be ascertained and then separated from the measurement.Preferably, near-normal incidence geometry includes angles that range±10 degrees from the normal, or ±5 degrees from the normal, or ±2degrees from the normal. In those circumstances in which the setuppermits, near-normal incidence geometry will include setups that have anormal incidence angle. We define reflectivity constants r_(R) for rightcircularly-polarized light component (analogous to constant r_(p) forp-polarization component) and r_(L) for left circularly-polarized lightcomponent (analogous to constant r_(s) for s-polarization component) andlet R_(N) be the magnitude of the right circularly-polarized componentand L_(N) be the magnitude of the left circularly-polarized component.In the embodiments practiced with Class II materials, the phase-shifteris one that works with circularly-polarized light. In preferredembodiments the phase-shifter is a Faraday rotator or any material thatin transmission or reflection mode can be used to add a relative phasedifference to the R and L components of the polarized light. As with theClass I embodiments, for Class II material embodiments, nulling of theodd harmonics is carried out by adjusting the phase-shifter (e.g.,Faraday rotator) so that the magnitude of the odd harmonic is minimized.In the Class II embodiments, a quarter-wave plate is used in conjunctionwith a conventional polarization analyzer. A quarter-wave plate is usedbecause it converts the two circularly-polarized components into twoorthogonal, linearly-polarized components without disturbing themagnitude and relative phase of the components. With Class IIembodiments, nulling of the even harmonic is carried out by adjustingthe orientation of the analyzer's transmission axis with respect to thequarter-wave plate so that the magnitude of the even harmonic isminimized.

With the analogies between Class I and Class II embodiments having beendescribed above, we now describe six setups for practicing Class IIembodiments of the invention.

In the first Class II setup, the initial beam has an ellipticalpolarization with components R₁≠L₁. The initial beam passes through apolarization modulator and emerges with components R₂ and L₂ with thenecessary condition that$\frac{R_{1}}{L_{1}} \neq {\frac{R_{2}}{L_{2}}.}$The beam then passes through a phase-shifter for circularly-polarizedlight, interacts with the sample, passes through a quarter-wave platethat converts two circularly-polarized components into two orthogonal,linearly-polarized components, then through the analyzer, and finallyfalls on the detector.

The second Class II setup has an initial elliptically polarized beamwith components R₁≠L₁. The initial beam passes through a polarizationmodulator and emerges with components R₂ and L₂ with the necessarycondition that $\frac{R_{1}}{L_{1}} \neq {\frac{R_{2}}{L_{2}}.}$The beam then interacts, in sequence, with the sample, a phase-shifterfor circularly-polarized light, passes through a quarter-wave plate thatconverts two circularly-polarized components into two orthogonal,linearly-polarized components, then through the analyzer, and finallyfalls on the detector.

The third Class II setup uses an initial light beam with an arbitraryinitial polarization, passes the beam through an analyzer (requiring, ofcourse, that a sufficient amount of light pass through the analyzer toprovide adequate signal-to-noise ratios for practice of the methods ofthe invention), then through a first quarter-wave plate, interacts withthe sample, then with the phase-shifter for circularly-polarized lightand emerges with resultant components R₁ and L₁, then with thepolarization modulator and emerges with components R₂ and L₂ with therequirement that $\frac{R_{1}}{L_{1}} \neq {\frac{R_{2}}{L_{2}}.}$The beam then passes through a second quarter-wave plate that convertstwo circularly-polarized components into two orthogonal,linearly-polarized components, followed by a polarizer with thetransmission axis oriented not parallel to either the slow axis (SA) orthe fast axis (FA) of the second quarter-wave plate, and finally fallson the detector.

The fourth Class II setup uses an initial light beam with an arbitraryinitial polarization, passes the beam through an analyzer (requiring, ofcourse, that a sufficient amount of light pass through the analyzer toprovide adequate signal to noise for practice of the methods of theinvention), then through a first quarter-wave plate, followed by aphase-shifter for circularly-polarized light, after which the beaminteracts with the sample, and emerges with resultant components R₁ andL₁. The beam then interacts with the polarization modulator and emergeswith components R₂ and L₂ with the requirement that$\frac{R_{1}}{L_{1}} \neq {\frac{R_{2}}{L_{2}}.}$The beam then passes through a second quarter-wave plate that convertstwo circularly-polarized components into two orthogonal,linearly-polarized components, followed by a polarizer with thetransmission axis oriented not parallel to either the slow axis (SA) orthe fast axis (FA) of the second quarter-wave plate, and finally fallson the detector.

In the fifth Class II setup, the initial beam has an ellipticalpolarization, it interacts with a device that variably changes therelative magnitude of the right-circular polarization component, R, tothe left-circular polarization component, L, for example, a circularlydichroic material with a variable thickness in transmission mode; such amaterial absorbs or transmits (as a result of absorption) theleft-circularly polarized light and the right-circularly polarized lightdifferently, and serves the same purpose as the analyzer quarter-waveplate combination described above, interacts with the sample, and thenwith a phase-shifter for circularly-polarized light and emerges withresultant components R₁ and L₁. The beam then interacts with thepolarization modulator and emerges with components R₂ and L₂ with therequirement that $\frac{R_{1}}{L_{1}} \neq {\frac{R_{2}}{L_{2}}.}$The beam then passes through a quarter-wave plate that converts twocircularly-polarized components into two orthogonal, linearly-polarizedcomponents, followed by a polarizer with the transmission axis orientednot parallel to either the slow axis (SA) or the fast axis (FA) of thequarter-wave plate, and finally falls on the detector.

In the sixth Class II setup, the initial beam has an ellipticalpolarization, it interacts with a device that variably changes therelative magnitude of the right-circular polarization component, R, tothe left-circular polarization component, L, (serving the same purposeas the analyzer quarter-wave plate combination described above),interacts with the phase-shifter for circularly-polarized light, andthen with the sample, and emerges with resultant components R₁ and L₁.The beam then interacts with the polarization modulator and emerges withcomponents R₂ and L₂ with the requirement that$\frac{R_{1}}{L_{1}} \neq {\frac{R_{2}}{L_{2}}.}$The beam then passes through a quarter-wave plate that converts twocircularly-polarized components into two orthogonal, linearly-polarizedcomponents, followed by a polarizer with the transmission axis orientednot parallel to either the slow axis (SA) or the fast axis (FA) of thequarter-wave plate, and finally falls on the detector.

EXAMPLE 1 Characterization of Nucleic Acid Hybridization on a Microarray

Hybridized microarrays were prepared as follows. DNA fragments of 0.3 Kblength were synthesized by the polymerase chain reaction. An AffymetrixGMS-417 microarrayer (Affymetrix, Inc., Santa Clara, Calif.) was usedaccording to the manufacturer's instructions to print an array on apoly-L-lysine treated glass slide having dimensions of 1.2 cm by 1.2 cm.The features are round spots having an average diameter of 100 micronsand an average center-to-center separation of 180 microns. The printedDNA fragments were cross-linked to the poly-L-lysine coated glasssurface by exposure to ultraviolet light. Approximately 1 pico-gram ofDNA is printed per spot, giving rise to a density within a spot on theorder of approximately 1×10¹³ molecules/cm². The glass slide was thendried in vacuo. Non-specific binding sites were blocked by treating theslide with succinic anyhydride in a sodium borate solution. These stepswere carried out following protocols published by Operon Technologies,Inc. “Protocol for Preparation of PLL Slides, Microarrays” (2001),“Procedure for Printing Slides, Microarrays” (2001), “Post ProcessingProtocol, Microarrays” (2001), Qiagen Operon N.V., incorporated hereinby reference.

Prior to hybridization, the DNA fragments on the slide were denatured byimmersing the slide in boiling water, followed by an immersion in 95%ethanol. The array was hybridized with Cy3-labeled cDNAs reversetranscribed from isolated RNA following the Operon Technologies, Inc.“Reverse Transcription, Microarrays” (2001) protocol, incorporatedherein by reference. The labeled cDNA molecules are on the order of 0.5Kb length. The Cy3-labeled cDNA was denatured and hybridized with thearray at 42° C. overnight under standard conditions set forth in theOperon Technologies, Inc. “Hybridization Protocol, Microarrays” (2001)protocol, incorporated herein by reference. Following the hybridizationstep, the array was washed in 2×SSC, 0.2% SDS for approximately 1minute, then in 0.2×SSC, and then in 0.05×SSC, following the OperonTechnologies, Inc. “Hybridization Protocol, Microarrays” (2001) protocolto remove non-hybridized cDNA. The slide was dried prior to carrying outoptical scanning, as described below.

Cy3 is a fluorescent dye having an absorption maximum at 550 nm and anemission maximum at 570 nm. The Cy3 dye attached to the hybridized cDNAwas photo-bleached using a Spectra-Physics diode-pumped solid-statelaser DPSS-532 (Spectra-Physics, Mountain View, Calif.) that delivers acontinuous power of 100 milliwatts at 532 nm. The fluorescence-labeledfeatures were exposed to a flux of 100 J/cm² (3×10²⁰ photons/cm²) whichreduced the fluorescence yield of the dye by about a factor of 10⁵.

For an oblique-incidence reflectance-difference scan as illustrated inFIG. 1, we use a 5 milli-Watt linearly-polarized He—Ne laser (modelLGK7654-7, LASOS, Ebersberg, Germany). The initial polarization is setalong the s-polarization direction by passing the beam (102) through apolarizer (103) (model 5524, New Focus, San Jose, Calif.). The beam ispassed through a photoelastic modulator (104) (model PEM90, HindsInstruments, Hillsboro, Oreg.) with the initial polarization bisectingthe two principal axes of the modulator. Periodically at Ω=50 kHz, thepolarization modulator produces an output polarization that is changedin a sequence of the original s-polarization, left circularpolarization, p-polarization, left circular polarization,s-polarization, right circular polarization, p-polarization, rightcircular polarization, and finally back to s-polarization. Thepolarization-modulated beam is then passed through a phase-shifter(105), in this instance, a Pockel cell (model Impact10, ClevelandCrystals, Highland Heights, Ohio) with the principal axes of the cellaligned parallel to s- and p-polarizations. The Pockel cell adds betweenthe s -and p -polarized components a phase lag, φ, that can be changedby applying an electrical voltage. The phase-lagged,polarization-modulated beam is focused through a lens (106) on a regionof the slide having no DNA features. The reflected beam (107) is passedthrough a lens (108) and an analyzing polarizer (i.e., analyzer) (109)(model 5524, New Focus, San Jose, Calif.) with its transmission axis(TA) set an angle of θ_(PL) with respect to the s-polarization axis. Theresultant beam with the intensity IR(t) is then detected with a biasedsilicon photodiode (110) (model 818-BB-40, Newport, Irvine, Calif.). Thephoto-current, in proportion to I_(R)(t), is sent to two Lock-inamplifiers (model SR830, Stanford Research Systems, Sunnyvale, Calif.)to detect the first and second harmonics in the photo-current, orequivalently, I_(R)(t).

We null (i.e., minimize the intensity of) the first harmonic byadjusting the voltage on the Pockel cell (105) and null the secondharmonic by adjusting the orientation of the transmission axis of theanalyzer (109). To obtain Im{Δ_(p)−Δ_(s)}, we measure the factorc₁(N=1)I_(inc)|r_(p0) sin θ_(PL)|² by adjusting the voltage on Pockelcell (105) until the detected first harmonic signal is maximized. Wedivide the subsequently measured first harmonic signal (i.e., after theinitial nulling) by this maximum signal to deduce Im{Δ_(p)−Δ_(s)}directly.

During the scan, the illuminating optics and detection system werefixed, and the chip was translated by mounting it on top of twoorthogonal translation stages (Newport-Klinger, model 462-XY, Irvine,Calif.). 80-pitch screws were attached to each of the stages and thetranslation stages were moved sequentially by turning the screws withstepper motors. The smallest achievable step size along both the x and ydirection on the surface was 0.8 microns in this experiment. To generatethe scan shown as FIG. 2, a 2.4 micron step size was used for x and y,as the size of the features in the array are on the order ofapproximately 40 steps or so (i.e. around 100 microns). FIG. 2 shows aplot of Im{Δ_(p)−Δ_(s)} which was obtained as described above.

The plot of Im{Δ_(p)−Δ_(s)} in FIG. 2 corresponds to refractive indexchanges that allow surface density variations of the features to beobserved, which in turn, can be correlated to the extent ofhybridization. The gray-scale variation reflects surface density changesfrom feature to feature. The lightest patch is the least hybridizedsingle-stranded DNA patch, while the darkest patch is the mostextensively hybridized. Also plainly visible is the feature-to-featurevariation present in the array, which could significantly influence theinterpretation of fluorescence yield measurements.

EXAMPLE 2 Characterization of Nucleic Acid Hybridization on aTwo-Dimensional Microarray

An Affymetrix GMS-417 microarrayer (Affymetrix, Inc., Santa Clara,Calif.) was used according to the manufacturer's instructions to preparearrays of features using a variety of different 60-mer DNAoligonucleotides (“60-mer oligos”) purchased from Sigma-Genosys (TheWoodlands, Tex.). The sequences of the 60-mers were designed so that asubset was predicted to hybridize with the population of cDNAs used inthe hybridization experiment. The 60-mer oligos were printed onpoly-L-lysine coated glass slides using the GMS-417 microarrayer, andwere then cross-linked to the poly-L-lysine coated glass surface byexposure to ultraviolet light. The slides subsequently were processedwith succinic anyhydride in a sodium borate solution to block outnon-specific binding sites. The printing, cross-linking, andpost-processing steps were carried out following protocols published byOperon Technologies, Inc. “Protocol for Preparation of PLL Slides,Microarrays” (2001), “Procedure for Printing Slides, Microarrays”(2001), “Post Processing Protocol, Microarrays” (2001), Qiagen OperonN.V., incorporated herein by reference. The printed features arecircularly shaped with a bi-lobular fine structure, have diameters of130 microns and an average center-to-center separation of 190 microns.The bi-lobular structure results from a shape modification of theGMS-417 transfer tip.

The array was subsequently hybridized with Cy5-labeled cDNAs reversetranscribed from isolated RNA following the Operon Technologies, Inc.“Reverse Transcription, Microarrays” (2001) protocol, incorporatedherein by reference. The labeled cDNA molecules are on the order of 0.5Kb length. The Cy5-labeled cDNA was first heat denatured. The denatured,Cy5-labeled cDNA molecules were then allowed to hybridize with the arrayon the slides at 67° C. for 12 hours or overnight under standardconditions set forth in the Operon Technologies, Inc. “HybridizationProtocol, Microarrays” (2001) protocol, incorporated herein byreference. Following the hybridization step, the array was washed in2×SSC, 0.2% SDS for approximately 1 minute, then in 0.2×SSC, and then in0.05×SSC, following the Operon Technologies, Inc. “HybridizationProtocol, Microarrays” (2001) protocol to remove non-hybridized cDNA.The slide was dried prior to carrying out optical scanning, as describedbelow.

Cy5 is a fluorescent dye having an optical absorption maximum at 649 nmand an emission maximum at 670 nm. To demonstrate that the approach ofthis invention is just as effective even when the optical labelingmolecules are not fluorescent, the Cy5 label on the hybridized cDNA wasphoto-bleached until no measurable fluorescence could be detected with astandard commercial fluorescence scanner.

For an oblique-incidence reflectivity-difference scan as illustrated inFIG. 1, we use a 5 milli-Watt linearly-polarized He—Ne laser (modelLGK7654-7, LASOS, Ebersberg, Germany). The initial polarization, set bypassing incident beam (102) through a polarizer (model 5524, New Focus,San Jose, Calif.) (103) is along the s-polarization direction. Theincident beam (102) is passed through a photoelastic modulator (104)(model PEM90, Hinds Instruments, Hillsboro, Oreg.) with the twoprincipal axes of the modulator bisecting the initial polarization.Periodically at Ω=50 kHz, the polarization modulator produces an outputpolarization that is changed in a sequence of the originallys-polarization, left-circular polarization, p-polarization,left-circular polarization, s-polarization, right-circular polarization,p-polarization, right-circular polarization, and finally back tos-polarization. The polarization-modulated beam is then passed through aphase-shifter (105), in this instance, a Pockel cell (model Impact10,Cleveland Crystals, Highland Heights, Ohio) with the principal axes ofthe cell aligned parallel to s- and p-polarizations. The Pockel celladds between the s- and p-polarized components a phase lag, φ, that canbe changed by applying an electrical voltage. The phase-lagged,polarization-modulated beam is focused by a lens (106) on a region ofthe slide having no DNA features. The reflected beam (107) is passedthrough another lens (108) and then through an analyzing polarizer (109)(model 5524, New Focus, San Jose, Calif.) with its transmission axis(TA) set an angle of θ_(PL) with respect to the s-polarization axis. Theresultant beam with the intensity I_(R)(t) is then detected with abiased silicon photodiode (110) (model-818-BB-40, Newport, Irvine,Calif.). The photo-current, in proportion to I_(R)(t), is sent to twoLock-in amplifiers (model SR830, Stanford Research Systems, Sunnyvale,Calif.) to detect the first and second harmonics in the photo-current,or equivalently, I_(R)(t).

We null (i.e., minimize the intensity of) the first harmonic byadjusting the voltage on the Pockel cell (105) and null the secondharmonic by adjusting the orientation of the transmission axis of theanalyzer (109). To obtain Im{Δ_(p)−Δ_(s)}, we measure the factorc₁(N=1)I_(inc)|r_(p0) sin θ_(PL)|² by adjusting the voltage on Pockelcell (105) until the detected first harmonic signal is maximized. Wedivide the subsequently measured first harmonic signal (i.e., after theinitial nulling) by this maximum signal to deduce Im{Δ_(p)−Δ_(s)}directly. To obtain Re{Δ_(p)−Δ_(s)}, we measure the factorc₂(N=2)I_(inc) |r_(p0) sin θ_(PL)|² by inserting another polarizinganalyzer either right before the optical beam is incident on the slidesurface or immediately after the optical beam is reflected off the slidesurface with the transmission axis oriented along either thes-polarization or the p-polarization direction, and measure the secondharmonic signal. We divide the subsequently measured second harmonicsignal (i.e., after the initial nulling) by two times the secondharmonic signal (obtained following insertion of the s-polarization orp-polarization oriented analyzer), and then multiply the result by theanalyzer transmittance for the passing polarization to deduceRe{Δ_(p)−Δ_(s)}.

During the scan, the illuminating optics and detection system werefixed, and the chip was translated by mounting it on top of twoorthogonal translation stages (Newport-Klinger, model 462-XY, Irvine,Calif.). 80-pitch screws were attached to each of the stages and thetranslation stages were moved sequentially by turning the screws withstepper motors. The smallest achievable step size along both the x and ydirection on the surface was 0.8 microns in this experiment. To generatethe scan shown as FIG. 3(a) through FIG. 3(d), step sizes of 4.8 micronwere used for x and y.

FIG. 3(a) shows a plot of Im{Δ_(p)−Δ_(s)}, corresponding to changes inrefractive index, over a region that contains a square array of 5×5=25printed features. 9 of the 25 distinct 60-mer oligos have hybridizedwith the Cy5-labeled cDNA molecules. Peaks 301, 302, 304, 308, 310, 314,315, 324, and 325 shown in FIG. 3(a) correspond to hybridized features.Other features with much smaller and yet well distinguished lobes (i.e.,peaks 303, 305-307, 309, 311-313, and 316-323) are those features thatdid not hybridize with the Cy5-labeled cDNA. On average, the peakheights for the hybridized features [Im{Δ_(p)−Δ_(s)}˜0.05] are 50 timesthe peak heights of those features that did not hybridize with theCy5-labeled cDNA [Im{Δ_(p)−Δ_(s)}˜0.001].

FIG. 3(b) shows a plot of Re{Δ_(p)−Δ_(s)}, corresponding to changes inoptical absorption coefficient, for the same array region as shown inFIG. 3(a). The changes in optical absorption coefficients arise from thepresence of the Cy5 dye in the labeled cDNA. Unlike Im{Δ_(p)−Δ_(s)}(corresponding to refractive index changes, allowing even the unlabeled60-mers to be imaged), for Re{Δ_(p)−Δ_(s)}, only those features wheresuccessful hybridization has taken place [i.e., FIG. 3(b) peaks 301,302, 304, 308, 310, 314, 315, 324 and 325] exhibit prominent peakscorresponding to the same peaks in FIG. 3(a). The maximum value ofRe{Δ_(p)−Δ_(s)} is ˜0.064 (peak 315). As expected, there are nodetectable peaks in Re {Δ_(p)−Δ_(s)} for array features that did nothybridize to the labeled cDNA.

FIG. 3(c) shows a plot of Im{Δ_(p)−Δ_(s)} (corresponding to refractiveindex changes) for another region of the slide containing a differentarray of features. This array was printed using a different set of 25distinct 60-mer oligos whose sequences were designed so as to not becomplementary to any of the labeled cDNA sequences. As expected, none ofthe features in this array hybridized with the sample cDNA molecules.Because the Im{Δ_(p)−Δ_(s)} scan tracks refractive index changes,features corresponding to the unlabeled 60-mer oligos are clearlydetected. Each spot yet again consists of two lobes. The averageIm{Δ_(p)−Δ_(s)} peak value of the lobes is ˜0.001. The scale of FIG.3(c) is expanded from that of FIG. 3(a) by a factor of 25.

FIG. 3(d) shows a plot of Re{Δ_(p)−Δ_(s)}, corresponding to changes inoptical absorption coefficient, for the same array region shown in FIG.3(c). The scale of FIG. 3(d) is expanded from that of FIG. 3(b) by afactor of 6. Because no hybridization took place in this array, weexpect no optically absorbing materials to be present in these arrayfeatures. This is indeed the case, as no discernable features areobservable above the background noise [FIG. 3(d)]. In contrast, FIG.3(c) shows that the molecular densities and the resultant refractiveindices inside these features are significantly different from those ofthe surrounding region. This means that even without labeling agents,the methods of the invention allow monitoring and measurement of densityvariations of the features on a microarray slide. See Appendix SpecialCase 4.

The cDNA molecules that hybridized with some of the 60-mer oligos on thearray contain photo-bleached Cy5 molecules that are still stronglyabsorbing at the 633 nm wavelength of our probe laser (near the 647 nmCy5 absorption peak). The strong absorption gives rise to large changesin the refractive index and optical absorption coefficient within thehybridized features. This is reflected in the 50-fold changes inIm{Δ_(p)−Δ_(s)} from the unhybridized to hybridized features [FIG.3(a)], and in similar changes in Re{Δ_(p)−Δ_(s)} [FIG. 3(b)]. Note thereis no detectable fluorescence yield from these photo-bleached Cy5molecules.

Since the large changes in the refractive index and optical absorptioncoefficient are highly wavelength-dependent, one may readily use twooptical labeling agents with the absorption maxima peaking at twosufficiently separated wavelengths for two sets of cDNA molecules. Indoing so and by scanning the features with two appropriate lasers as iscommon practice in fluorescence scanners, the difference in the degreeof hybridization between the two sets can be determined and quantifiedwithout measurement of fluorescence yield and the difficulties attendingsuch measurement.

EXAMPLE 3 Characterization of Xenon Monolayer Growth on Nb (110)

As a model system for heteroepitaxy on a highly mismatched substrate, westudied the vapor-phase epitaxy of Xe on Nb(110) from 33 K to 100 Kusing a combination of low energy electron diffraction (LEED) and thein-situ oblique-incidence reflectivity difference approach of theinvention. We use the same experimental setup as shown in FIG. 1 and thesame optical components as described in Example 1 except that thephase-lagged, polarization-modulated beam remains collimated when itilluminates and subsequently reflects off the clean and Xe covered Nb(110) surface.

Let r_(p0)=|r_(p0)|exp(iΦ_(p0)) and r_(s0)=|r_(s0)|exp(iΦ_(s0)) be thereflection coefficients for p- and s-polarized light at λ=6328 Å(LGK7654-7 He—Ne laser wavelength) from a clean Nb(110) substrate beforedeposition of xenon. Let r_(p)=|r_(p)|exp(iΦ_(p)) andr_(s)=|r_(s)|exp(iΦ_(s)) be the reflection coefficients for p- ands-polarized light coefficients after a Xe layer is deposited on thesubstrate. As above, we define Δ_(p)=(r_(p)−r_(p0))/r_(p0) andΔ_(s)=(r_(s)−r_(s0))/r_(s0). Before deposition, we null the firstharmonic by adjusting the phase lag introduced by the Pockel cell andnull the second harmonic by adjusting the orientation of the analyzertransmission axis with respect to the s-polarization axis. Subsequentlywe expose the clean Nb(110) surface to an ambient of xenon gas at1.4×10⁻⁷ Torr. The change in the first harmonic is converted toIm{Δ_(p)−Δ_(s)} by using the same procedure described in Example 2. Thechange in the second harmonic is converted to Re{Δ_(p)−Δ_(s)} using thesame procedure described in Example 2. FIG. 4 shows on the ordinateIm{Δ_(p)−Δ_(s)} (corresponding to refractive index changes) vs. thedeposition time (abscissa) during the Xe growth on Nb(110) from 40 K to60 K. Two plateaus (401, 402) mark completion of first and secondmonolayers of xenon on Nb(110).

FIG. 5 shows on the ordinate Re{Δ_(p)−Δ_(s)} (corresponding to opticalabsorption coefficient changes) vs. deposition time (abscissa) duringthe Xe growth on Nb(110) at 40 K. The difference (502) between theoriginal signal (501) and the envelope part of the signal (not shown)reveals peaks that are roughly periodic with the deposition time. Thepeaks are indicative of an incomplete layer-by-layer growth that isdominated by a step-flow growth at this temperature. Arrowhead 503 markscompletion of two monolayers. Left ordinate is Re{Δ_(p)−Δ_(s)}, rightordinate is variation in Re{Δ_(p)−Δ_(s)}.

EXAMPLE 4 Characterization and Control of Lead (Pb) growth on Cu (100)in Electrochemical Deposition Environment

We have studied the electrodeposition of monolayers of lead (Pb) on aCu(100) surface using the methods of the present invention. This exampleillustrates application of the methods for characterizing andcontrolling in situ the growth and removal of a thin film of a thirdmaterial at the interface between two materials, one of which being aliquid and the other is a solid. In electrodeposition, the growth of thethird material at the interface is governed by the chemical make-up ofthe liquid containing the third material and the electrostatic potentialdifference between the liquid and the solid material.

FIG. 6(a) shows Re{Δ_(p)−Δ_(s)} (602) vs. time during Pb growth onCu(100) at room temperature as the potential is changed in a step-wisemanner. FIG. 6(b) shows the corresponding Im{Δ_(p)−Δ_(s)} (607). Whenthe potential (601) is initially stepped down from −0.222 V to −0.535 V,Re{Δ_(p)−Δ_(s)} (602) increases from zero to 0.02 at which point theslope of the increase significantly decreases. This crossover point(603) marks the deposition of the first monolayer of Pb on Cu. Thedotted horizontal line (606) marks the signal level in response to thedeposition of the first Pb monolayer in the form of an alloy. Themagnitude of the increase (˜0.02) indicates that the first Pb monolayerforms an alloy with the topmost Cu layer. After roughly 20 seconds whena transition layer is formed on top of the first monolayer, a bulk-like,compact Pb film starts to grow as indicated by a linear decrease inRe{Δ_(p)−Δ_(s)} (604) and Im{Δ_(p)−Δ_(s)} (605). When the potential isstepped back to −0.48 V at which only the first Pb monolayer is stableon Cu(100), the Pb overlayer on top of the first Pb monolayer (in alloyform) starts to dissolve into the liquid. CorrespondinglyRe{Δ_(p)−Δ_(s)} and Im{Δ_(p)−Δ_(s)} change in a backward manner almostalong their original paths until the values corresponding to the firstmonolayer are reached. When the potential is increased again from −0.48V to −0.222 V, the first Pb monolayer in form of alloy also dissolves inthe solution and the Cu(100) is essentially restored. The quantitativedifference in optical signals for growth and dissolution are due to theirreversibility in the film morphology evolution as different kineticsare predominant during growth and dissolution.

As FIGS. 6(a) and (b) illustrate, the thickness and morphology of thethin film of said third material or the electrodeposition process ingeneral can be controlled in situ by controlling the potential as afunction of time using the optical signals generated by the methods ofthe invention as feedback parameters.

Appendix

Let r_(p0)=|r_(p0)|exp(iΦ_(p0)) and r_(s0)=|r_(s0|exp(iΦ) _(s0))respectively be the reflection coefficients for p- and s-polarized lightfrom a surface. Let the optical dielectric constant of the material onthe incidence side be ε_(inc)≡(n_(inc)+iκ_(inc))², with n_(inc) andκ_(inc) being the corresponding refractive index and optical absorptioncoefficient. Let the optical dielectric constant of the material on thetransmission side be ε_(s)=(n_(s)+iκ_(s))², with n_(s) and κ_(s) beingthe corresponding refractive index and optical absorption coefficient.Let r_(p)=|r_(p)|exp(iΦ_(p)) and r_(s)=|r_(s)|exp(iΦ_(s)) respectivelybe the reflection coefficients for p- and s-polarized light from saidsurface when it is covered with a thin layer of a third material withrefractive index n_(d) and optical absorption coefficient κ_(d). Theoptical dielectric constant of said third material of interest isε_(d)=(n_(d)+i κ_(d))². The thickness of said third material is d. LetΔ_(p)=(r_(p)−r_(p0))/r_(p0) and Δ_(s)=(r_(s)−r_(s0))/r_(s0). It can beshown that: $\begin{matrix}{{\Delta_{p} - \Delta_{s}} = {{- {{\mathbb{i}}\left( \frac{4{\pi cos\theta}_{inc}\sin^{2}\theta_{inc}d}{\lambda} \right)}}\frac{\sqrt{ɛ_{inc}}{ɛ_{s}\left( {ɛ_{d} - ɛ_{inc}} \right)}\left( {ɛ_{d} - ɛ_{s}} \right)}{ɛ_{d}\left( {{ɛ_{s}^{2}\cos^{2}\theta_{inc}} - {ɛ_{inc}ɛ_{s}} + {ɛ_{inc}^{2}\sin^{2}\theta_{inc}}} \right)}}} & (3)\end{matrix}$where λ is the wavelength of the light beam in vacuo. θ_(inc) is theangle of incidence.General Case: Absorptive Materials on Both Sides of a Thin Layer of aThird Material (i.e., Im{ε_(s)}≠0, Im{ε_(inc)}≠0)

We measure both Re{Δ_(p)−Δ_(s)} and Im{Δ_(p)−Δ_(s)}. Knowing ε_(s) andε_(inc), we solve for ε_(d) as follows $\begin{matrix}{{ɛ_{d} = \frac{\left( {ɛ_{s} + ɛ_{inc} - A} \right) \pm \sqrt{\left( {ɛ_{s} + ɛ_{inc} - A} \right)^{2} - {4ɛ_{s}}}}{2}}{{Here}\quad{we}\quad{have}\quad{defined}}} & (4) \\{A = {{\mathbb{i}}\frac{\left( {\Delta_{p} - \Delta_{s}} \right){\lambda\left( {{ɛ_{s}^{2}\cos^{2}\theta_{inc}} - {ɛ_{inc}ɛ_{s}} + {ɛ_{inc}^{2}\sin^{2}\theta_{inc}}} \right)}}{4{\pi cos\theta}_{inc}\sin^{2}\theta_{inc}ɛ_{s}\sqrt{ɛ_{inc}}d}}} & (5)\end{matrix}$Special Case 1: Transparent Materials on Both Sides of a Thin Layer of aThird Material (i.e., Im{ε_(s)}=Im{ε_(inc)}=0)

From Equation (3), one can see that Re{Δ_(p)−Δ_(s)} is non-vanishingonly if said third material is absorptive, κ_(d)≠0. ConsequentlyRe{Δ_(p)−Δ_(s)} corresponds primarily to the optical absorptioncoefficient of said third material. When κ_(d)=0, Re{Δ_(p)−Δ_(s)}vanishes. In this case non-vanishing Im{Δ_(p)−Δ_(s)} corresponds to thedifference in the refractive indices of the third material and thematerials on both sides of the thin layer.

Special Case 2: Transparent Materials on Both Sides of a Thin Layer of aThird Material (i.e., Im{ε_(s)}=Im{ε_(inc)}=0), ε_(d) deviates slightlyfrom ε_(s)

In this case, Equation (3) can be simplified as $\begin{matrix}{{\Delta_{p} - \Delta_{s}} = {{- {{\mathbb{i}}\left\lbrack \frac{4\pi\quad d\quad\cos\quad\theta_{inc}\sin^{2}\theta_{inc}\sqrt{ɛ_{inc}}}{\lambda\left( {{ɛ_{s}\cos^{2}\theta_{inc}} - {ɛ_{inc}\sin^{2}\theta_{inc}}} \right)} \right\rbrack}}{\left( {ɛ_{d} - ɛ_{s}} \right).}}} & (6)\end{matrix}$Let the refractive index difference be δn_(d)=n_(d)−n_(s), we canrewrite Equation (6) as $\begin{matrix}{{\Delta_{p} - \Delta_{s}} = {\left\lbrack \frac{8\pi\quad d\quad\cos\quad\theta_{inc}\sin^{2}\theta_{inc}\sqrt{ɛ_{inc}}n_{s}}{\lambda\left( {{ɛ_{s}\cos^{2}\theta_{inc}} - {ɛ_{inc}\sin^{2}\theta_{inc}}} \right)} \right\rbrack{\left( {\kappa_{d} - {{\mathbb{i}}\quad\delta\quad n_{d}}} \right).}}} & (7)\end{matrix}$From Equation (7), one finds that Re{Δ_(p)−Δ_(s)} measures the opticalabsorption coefficient of a third material while Im{Δ_(p)−Δ_(s)}measures the difference in the refractive indices of the third materialand the material on the transmission side.Special Case 3: Transparent Materials on Both Sides (i.e.,Im{ε_(s)}=Im{ε_(inc)}=0), ε_(d) Deviates Slightly from ε_(inc)

In this case, Equation (3) can be simplified as $\begin{matrix}{{\Delta_{p} - \Delta_{s}} = {{+ {{\mathbb{i}}\left\lbrack \frac{4\pi\quad d\quad\cos\quad\theta_{inc}\sin^{2}\theta_{inc}ɛ_{s}}{\lambda\sqrt{ɛ_{inc}}\left( {{ɛ_{s}\cos^{2}\theta_{inc}} - {ɛ_{inc}\sin^{2}\theta_{inc}}} \right)} \right\rbrack}}\left( {ɛ_{d} - ɛ_{inc}} \right)}} & (8)\end{matrix}$Let the refractive index difference be δn_(d)=n_(d)−n_(inc), we canrewrite Equation (8) as $\begin{matrix}{{\Delta_{p} - \Delta_{s}} = {\left\lbrack \frac{8\pi\quad d\quad\cos\quad\theta_{inc}\sin^{2}\theta_{inc}ɛ_{s}}{\lambda\left( {{ɛ_{s}\cos^{2}\theta_{inc}} - {ɛ_{inc}\sin^{2}\theta_{inc}}} \right)} \right\rbrack\left( {{- \kappa_{d}} + {{\mathbb{i}}\quad\delta\quad n_{d}}} \right)}} & (9)\end{matrix}$From Equation (9), one finds that Re{Δ_(p)−Δ_(s)} measures the opticalabsorption coefficient of the third material while Im{Δ_(p)−Δ_(s)}measures the difference in the refractive indices of said third materialand the material on the incidence side.Special Case 4: Transparent Materials on Both Sides (i.e.,Im{ε_(s)}=Im{ε_(inc)}=0), ε_(d) Deviates Slightly from ε_(inc)=1, and aThin Layer of a Third Material that Consists of Molecules

In this case,ε_(d)−1≅4πγ_(mol) N _(s)(x, y;t)/d   (10)where γ_(mol) is the molecular polarizability of said third material,N_(s)(x,y;t) is the surface density of the molecules that may vary asfunctions of spatial coordinates (x and y) along the interface and/or asa function of time (t), d is the thickness of the thin layer.Consequently Equation (8) can be simplified as $\begin{matrix}{{\Delta_{p} - \Delta_{s}} = {{+ {{\mathbb{i}}\left\lbrack \frac{16{\pi\quad}^{2}\quad\cos\quad\theta_{inc}\sin^{2}\theta_{inc}ɛ_{s}\gamma_{mol}}{\lambda\left( {{ɛ_{s}\cos^{2}\theta_{inc}} - {\sin^{2}\theta_{inc}}} \right)} \right\rbrack}}{N_{s}\left( {x,{y\quad;t}} \right)}}} & (11)\end{matrix}$From Equation (11), one can measure the surface density differences asresults of displacements in spatial coordinates and/or in time from themeasurements of Im{Δ_(p)−Δ_(s)} and Re{Δ_(p)−Δ_(s)}, independent of thethickness of the layer. In cases when the molecular polarizabilityγ_(mol) is known or can be ascertained, one can determine the surfacedensity N_(s)(x,y;t) quantitatively from Im{Δ_(p)−Δ_(s)} andRe{Δ_(p)−Δ_(s)} using Eq. (11).

1-81. (canceled)
 82. An optical device, comprising: a sample holder; alight source for emitting a polarized incident light beam, saidpolarized incident light beam forming an incidence angle with a sampleheld by said sample holder; a polarization modulator that modulates at amodulation frequency the polarization of at least one of said incidentlight beam or a light beam reflected off a sample held by said sampleholder; an adjustable phase shifter located within said incident lightbeam path or said reflected light beam path; a detector located withinsaid reflected light beam path, wherein said detector is capable oftransducing said reflected light beam intensity to a variable electricalsignal; and a signal analyzer for receiving said variable electricalsignal from said detector and determining a magnitude of a frequencycomponent corresponding to a harmonic frequency of said modulationfrequency.
 83. The device of claim 82, wherein said incidence angle isoblique.
 84. The device of claim 83, wherein said polarization modulatoris selected from the group consisting of a photoelastic modulator, anelectro-optic phase modulator, a rotatable wave plate, a rotatabledouble-Fresnel rhomb, a rotatable reflecting surface, and a pair ofrotatable reflecting surfaces.
 85. The device of claim 83, wherein saidadjustable phase shifter is selected from the group consisting of avoltage-controlled Pockel cell, a mechanically controlled Soleil-Babinetcompensator, a Berek compensator, and a double-Fresnel rhomb achromaticretarder.
 86. The device of claim 83, wherein said adjustable phaseshifter is located within said incident light beam path.
 87. The deviceof claim 83, wherein said adjustable phase shifter is located withinsaid reflected light beam path.
 88. The device of claim 83, furthercomprising a polarizer or an analyzer located within said reflectedlight beam path between said sample holder and said detector.
 89. Thedevice of claim 88, comprising said polarizer.
 90. The device of claim88, comprising said analyzer.
 91. The device of claim 82, wherein saidincidence angle is near-normal.
 92. The device of claim 91, wherein saidpolarization modulator is selected from the group consisting of aphotoelastic modulator, an electro-optic phase modulator, a rotatablewave plate, a rotatable double-Fresnel rhomb, a rotatable reflectingsurface, and a pair of rotatable reflecting surfaces.
 93. The device ofclaim 91, wherein said adjustable phase shifter is selected from thegroup consisting of a Faraday rotator, and a material capable of addinga relative phase difference to the R and the L components of acircularly-polarized light beam.
 94. The device of claim 91, whereinsaid adjustable phase shifter is located within said incident light beampath.
 95. The device of claim 91, wherein said adjustable phase shifteris located within said reflected light beam path.
 96. The device ofclaim 91, further comprising a wave plate and a polarizer or a waveplate and an analyzer located within said reflected light beam pathbetween said sample holder and said detector.
 97. The device of claim96, comprising said wave plate and said polarizer.
 98. The device ofclaim 96, comprising said wave plate and said analyzer.
 99. An opticaldevice, comprising: a sample holder; a light source for emitting apolarized incident light beam, said polarized incident light beamforming an incidence angle with a sample held by said sample holder; apolarization modulator that modulates at a modulation frequency thepolarization of at least one of said incident light beam or a light beamreflected off a sample held by said sample holder; an analyzer or acombination of a wave plate and an analyzer located within said incidentlight beam path or said reflected light beam path, wherein theorientation of said analyzer transmission axis, θ_(PL), is adjustable; adetector located within said reflected light beam path, wherein saiddetector is capable of transducing said reflected light beam intensityto a variable electrical signal; and a signal analyzer for receivingsaid variable electrical signal from said detector and determining amagnitude of a frequency component corresponding to a harmonic frequencyof said modulation frequency.
 100. The device of claim 99, wherein saidincidence angle is oblique.
 101. The device of claim 100, wherein saidpolarization modulator is selected from the group consisting of aphotoelastic modulator, an electro-optic phase modulator, a rotatablewave plate, a rotatable double-Fresnel rhomb, a rotatable reflectingsurface, and a pair of rotatable reflecting surfaces.
 102. The device ofclaim 100, further comprising a polarizer located within said reflectedlight beam path between said sample holder and said detector.
 103. Thedevice of claim 99, wherein said incidence angle is near-normal. 104.The device of claim 103, wherein said polarization modulator is selectedfrom the group consisting of a photoelastic modulator, an electro-opticphase modulator, a rotatable wave plate, a rotatable double-Fresnelrhomb, a rotatable reflecting surface, and a pair of rotatablereflecting surfaces.
 105. The device of claim 103, wherein said waveplate and said analyzer are located within said incident light beampath, and further comprising a second wave plate and a polarizer locatedwithin said reflected light beam path between said sample holder andsaid detector.
 106. An optical device, comprising: a sample holder; alight source for emitting a polarized incident light beam, saidpolarized incident light beam forming a near-normal incidence angle witha sample held by said sample holder; a material within said incidentlight beam path capable of adding a relative magnitude difference to theR and the L components of said polarized incident light beam; apolarization modulator that modulates at a modulation frequency thepolarization of a light beam reflected off a sample held by said sampleholder; a wave plate and a polarizer located within said reflected lightbeam path, wherein the orientation of said polarizer transmission axis,θ_(PL), is adjustable; a detector located within said reflected lightbeam path, wherein said detector is capable of transducing saidreflected light beam intensity to a variable electrical signal; and asignal analyzer for receiving said variable electrical signal from saiddetector and determining a magnitude of a frequency componentcorresponding to a harmonic frequency of said modulation frequency.